smooth.construct.mpi.smooth.spec: Constructor for monotone increasing P-splines in SCAMs

Description

This is a special method function for creating smooths subject to a monotone increasing constraint which is built by the mgcv constructor function for smooth terms, smooth.construct. It is constructed using monotonic P-splines. This smooth is specified via model terms such as s(x,k,bs="mpi",m=2), where k denotes the basis dimension and m+1 is the order of the B-spline basis.

mpiBy.smooth.spec works similar to mpi.smooth.spec but without applying an identifiability constraint ('zero intercept' constraint). mpiBy.smooth.spec should be used when the smooth term has a numeric by variable that takes more than one value. In such cases, the smooth terms are fully identifiable without a 'zero intercept' constraint, so they are left unconstrained. This smooth is specified as s(x,by=z,bs="mpiBy"). See an example below.

However a factor by variable requires identifiability constraints, so s(x,by=fac,bs="mpi") is used in this case.

Usage

# S3 method for mpi.smooth.spec
smooth.construct(object, data, knots)
# S3 method for mpiBy.smooth.spec
smooth.construct(object, data, knots)

Value

An object of class "mpi.smooth", "mpiBy.smooth".

Arguments

object: A smooth specification object, generated by an s term in a GAM formula.
data: A data frame or list containing the data required by this term, with names given by object$term. The by variable is the last element.
knots: An optional list containing the knots supplied for basis setup. If it is NULL then the knot locations are generated automatically.

Author

Natalya Pya <nat.pya@gmail.com>

Details

The constructor is not called directly, but as with gam(mgcv) is used internally.

If the knots of the spline are not supplied, then they are placed evenly throughout the covariate values. If the knots are supplied, then the number of supplied knots should be k+m+2, and the range of the middle k-m knots must include all the covariate values.

References

Pya, N. and Wood, S.N. (2015) Shape constrained additive models. Statistics and Computing, 25(3), 543-559

Pya, N. (2010) Additive models with shape constraints. PhD thesis. University of Bath. Department of Mathematical Sciences

Examples

Run this code

## Monotone increasing P-splines example 
  ## simulating data...
   require(scam)
   set.seed(12)
   n <- 100
   x <- runif(n)*4-1
   f <- 4*exp(4*x)/(1+exp(4*x))
   y <- rpois(n,exp(f))
   dat <- data.frame(x=x,y=y)
 ## fit model ...
   b <- scam(y~s(x,k=15,bs="mpi"),family=poisson(link="log"),
       data=dat)
 ## fit unconstrained model...
   b1 <- scam(y~s(x,k=15,bs="ps"),family=poisson(link="log"),
         data=dat)
 ## plot results ...
   plot(x,y,xlab="x",ylab="y")
   x1 <- sort(x,index=TRUE)
   lines(x1$x,exp(f)[x1$ix])      ## the true function
   lines(x1$x,b$fitted.values[x1$ix],col=2)  ## monotone fit 
   lines(x1$x,b1$fitted.values[x1$ix],col=3) ## unconstrained fit 

## example with supplied knots...
   knots <- list(x=c (-1.5,  -1.2, -.99, -.97, -.7, -.5, -.3, 0, 0.7,  
           0.9,1.1, 1.22,1.5,2.2,2.77,2.93,2.99, 3.2,3.6))
   b2 <- scam(y~s(x,k=15,bs="mpi"),knots=knots, 
          family=poisson(link="log"), data=dat)
   summary(b2)
   plot(b2,shade=TRUE)

if (FALSE) {
## example with two terms...
   set.seed(0)
   n <- 200
   x1 <- runif(n)*6-3
   f1 <- 3*exp(-x1^2) # unconstrained term
   x2 <- runif(n)*4-1;
   f2 <- exp(4*x2)/(1+exp(4*x2)) # monotone increasing smooth
   f <- f1+f2
   y <- f+rnorm(n)*.7
   dat <- data.frame(x1=x1,x2=x2,y=y)
   knots <- list(x1=c(-4,-3.5,-2.99,-2.7,-2.5,-1.9,-1.1,-.9,-.3,0.3,.8,1.2,1.9,2.3,
2.7,2.99,3.5,4.1,4.5), x2=c(-1.5,-1.2,-1.1, -.89,-.69,-.5,-.3,0,0.7, 
0.9,1.1,1.22,1.5,2.2,2.77,2.99,3.1, 3.2,3.6))
   b3 <- scam(y~s(x1,k=15)+s(x2,bs="mpi", k=15), 
         knots=knots,data=dat)
   summary(b3)
   plot(b3,pages=1,shade=TRUE)
## setting knots for f(x2) only...
   knots <- list(x2=c(-1.5,-1.2,-1.1, -.89,-.69,-.5,-.3,
   0,0.7,0.9,1.1,1.22,1.5,2.2,2.77,2.99,3.1, 3.2,3.6))
   b4 <- scam(y~s(x1,k=15,bs="bs")+s(x2,bs="mpi",k=15),
       knots=knots,data=dat)
   summary(b4)
   plot(b4,pages=1,shade=TRUE)

## 'by' factor example... 
 set.seed(10)
 n <- 400
 x <- runif(n, 0, 1)
 ## all three smooths are increasing...
 f1 <- log(x *5) 
 f2 <-  exp(2*x) - 4
 f3 <-  5* sin(x)
 e <- rnorm(n, 0, 2)
 fac <- as.factor(sample(1:3,n,replace=TRUE))
 fac.1 <- as.numeric(fac==1)
 fac.2 <- as.numeric(fac==2)
 fac.3 <- as.numeric(fac==3)
 y <- f1*fac.1 + f2*fac.2 + f3*fac.3 + e 
 dat <- data.frame(y=y,x=x,fac=fac,f1=f1,f2=f2,f3=f3)
 b5 <- scam(y ~ fac+s(x,by=fac,bs="mpi"),data=dat)  
 plot(b5,pages=1,scale=0,shade=TRUE)
 summary(b5)
 vis.scam(b5,theta=50,color="terrain")

 ## comparing with unconstrained fit...
 b6 <- scam(y ~ fac+s(x,by=fac),data=dat) 
 x11()
 plot(b6,pages=1,scale=0,shade=TRUE)
 summary(b6)
 vis.scam(b6,theta=50,color="terrain")

 ## Note that since in scam() as in mgcv::gam() when using factor 'by' variables, 'centering'
 ## constraints are applied to the smooths, which usually means that the 'by'
 ## factor variable should be included as a parametric term, as well. 


## numeric 'by' variable example...
 set.seed(3)
 n <- 200
 x <- sort(runif(n,-1,2))
 z <- runif(n,-2,3)
 f <- exp(1.3*x)-5
 y <- f*z + rnorm(n)*2
 dat <- data.frame(x=x,y=y,z=z)
 b <- scam(y~s(x,by=z,bs="mpiBy"),data=dat)
 plot(b,shade=TRUE)
 summary(b)
 ## unconstrained fit...
 b1 <- scam(y~s(x,k=15,by=z),data=dat)
 plot(b1,shade=TRUE)
 summary(b1)
 }

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